The outward cusp domain with a polynomial type singularity is a typical example which is not a \((W^{1, p}, W^{1, p})\) -extension domain for \(1\le p<\infty \) . However, it is a \((W^{1, p}, W^{1, q})\) -extension domain, for some \(1\le q<p\le \infty \) , see the monograph [1]} and the references therein. Hence, for \(1\le q<p\le \infty \) , it is not necessary for a \((W^{1, p}, W^{1, q})\) -extension domain to be Ahlfors regular. In the absence of Ahlfors regularity, one has to find alternative approaches for proving \(|\partial {\Omega }|=0\) . The first approach in [2]}, [3]} was to generalize the Ahlfors regularity (REF ) to a Ahlfors-type estimate
\(|B(x, r)|^p\le C\Phi ^{p-q}(B(x, r))|B(x, r)\cap {\Omega }|^q\) 
